Digital Signal Processing Reference
In-Depth Information
A
0
≤
t
≤
T
(a)
x
1
(
t
)
=
−
A
−
T
≤
t
≤
0;
T
2
A
≤
t
≤
T
=
(b)
x
2
(
t
)
≤
t
≤
T
−
A
0
2
;
T
2
A
≤
t
≤
T
(c)
x
3
(
t
)
=
≤
t
≤
T
00
2
.
4.2
For the functions
φ
1
(
t
)
−
2
t
−
4
t
=
e
and
φ
2
(
t
)
=
1
−
K
e
determine the value of
K
such that the functions are orthogonal over the
interval [
−∞, ∞
].
4.3
The Legendre polynomials are widely used to approximate functions. An
n
th-order Legendre polynomial
P
n
(
x
) is defined as
=
M
d
n
d
x
n
(
x
2
−
1
n
!2
n
1)
n
a
nm
x
m
,
P
n
(
x
)
=
m
=
0
where the values of
a
nm
can be expressed as follows:
n
(
n
+
m
)!
2
n
m
!(
n
−
m
/
2)!(
n
+
m
/
2)!
1)
(
n
−
m
)
/
2
=
−
a
nm
(
m
=
0
n
,
m
odd
n
,
m
even
Note that
a
nm
is non-zero only when both
n
and
m
are either odd or even.
For all other values of
n
and
m
,
a
nm
is zero. The first few orders of Legendre
polynomials are given by
P
2
(
x
)
=
1
2
(3
x
2
−
1);
P
0
(
x
)
=
1;
=
1
2
(5
x
3
−
P
1
(
x
)
=
x
;
P
3
(
x
)
3
x
);
and are shown in Fig. P4.3.
The Legendre polynomials
{
P
n
(
x
)
,
n
=
0
,
1
,
2
,...}
form a set of
orthogonal functions over the interval [
−
1, 1] by satisfying the follow-
ing property:
1
2
2
m
+
m
=
n
P
m
(
x
)
P
n
(
x
)d
x
=
1
0
m
=
n
.
−
1
Verify the above orthogonality condition for
m
,
n
=
0, 1, 2, 3.
4.4
The Chebyshev polynomials of the first kind are used as the approximation
to a least-squares fit. The
n
th-order polynomial
T
n
(
x
) can be expressed as
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